text: In an aperiodic crystal the structure can no longer be simply described by three different vectors in real or reciprocal space. In general there is a substructure describable by three (e.g. ), similar to supercells above, but in addition there is some additional periodicity (one to three) which cannot be described as a multiple of the three; it is a genuine additional periodicity which is an irrational number relative to the subcell lattice. The diffraction pattern can then only be described by more than three indices.
  An extreme example of this is for quasicrystals, which can be described similarly by a higher number of Miller indices in reciprocal space—but not by any translational symmetry in real space. An example of this is shown in Figure 15 for an Al-Cu-Fe-Cr decagonal quasicrystal grown by magnetron sputtering on a sodium chloride substrate and then lifted off by dissolving the substrate with water. In the pattern there are pentagons which are a characteristic of the aperiodic nature of these materials.